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Abstract
X-ray grating-based techniques often lead to artifacts in the phase retrieval process of phase objects presenting very fast spatial transitions or sudden jumps, especially in the field of non-destructive testing and evaluation. In this paper, we present a method that prevents the emergence of artifacts by building an interferogram corrected from any variations of the object intensity and given as input in the phase retrieval process. For illustration, this method is applied to a carbon fiber specimen imaged by a microfocus X-ray tube and a single 2D grating. A significant reduction of artifacts has been obtained, by a factor higher than 10. This evaluation has been performed experimentally thanks to the Confidence Map tool, a recently developed method that estimates the error distribution from the phase gradient information.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
One of the most used techniques to measure the wavefront phase shift in X-rays consists in introducing a modulator between the source and the detector to produce a reference intensity pattern. This pattern becomes distorted when the wavefront is aberrant, which is the case after adding a phase object into the optical path. The reference intensity pattern can be randomly or regularly distributed: speckle-based techniques [1,2] use randomly distributed intensity patterns while grating-based techniques [3–10] use regularly distributed intensity patterns, that we call interferograms. Among the regular 2D grating-based devices, multi-lateral shearing interferometers (MLSI) simultaneously measure the wavefront phase shift in multiple orientations [11–13]. In X-ray phase imaging using MLSI, complex phase objects with abrupt intensity transitions such as edges or splits, can be strenuous to be phase sampled by the grating. It leads to under-sampling artifacts in the phase image when the transitions of the object evolve faster than the period of the interferogram. Classically, apodization techniques could be performed in order to reduce artifacts however by lowering the spatial resolution of the phase image [14]. To preserve the spatial resolution while reducing artifacts, another recent approach [15] proposed to combine several images acquired while spatially shifting the grating. However this method is only applicable in the case of low-acutance phase images. Here we propose a preventive approach that performs on high-acutance phase images which we call Method of Artifacts Reduction from the Intensity of the Object (MARIO). After introducing the problems in section 2. based on simulated data, we present the formalism of this new process in section 3, and its application on experimental data in section 4. Since the proposed technique is somewhat cumbersome, as it requires the acquisition of an additional image, we evaluate its benefits in a quantitative way by using the Confidence Map, a tool that we have introduced in our previous work [16].
2. Influence of abrupt phase and attenuation transition in the retrieval process: illustration using the MLSI technique
Based on a wavefront approach, simulated data presented in Fig. 1 (a,b) summarize the MLSI procedure. Using a monochromatic divergent point source at 17.48 keV (Mo K$\alpha$ line) with a detection plane made of $2048\times 2048$ squared pixels of $6.5\times 6.5\,{\mathrm {\mu }}$m$^{2}$, Fig. 1 (a) presents the image of a 2D-checkerboard phase grating with a $12\,{\mathrm {\mu }}$m orthogonal period and a $[0-\pi ]$ shift at Mo K$\alpha$. This image is called the reference interferogram $I_{ref}$. The second image, $I_{mod}$ (Fig. 1 (b)) is produced with the same grating parameters but adding a canonical object, here a 750 ${\mathrm {\mu }}$m radius ball made of PMMA material. Finally, Fig. 1 (c) presents the image of the PMMA ball $I_{obj}$ without the grating, produced with the same object parameters. The source-detector distance is $d_{sd}=60$ cm for a grating and an object magnification factor of $M_g=5$ and $M_o=7$ respectively. The underlying setup of the simulation study is presented in Fig. 5.Here, we suppose thin objects compared to the propagation distances. In this context, placing the grating upstream or downstream of the object does not influence the results. Since the MLSI uses a periodic modulator, phase retrieval is classically performed according to Fourier demodulation technique [17] and compares the phase shift between $I_{ref}$ and $I_{mod}$ by studying the spectrum variations induced by the object. All corresponding Fourier transforms modulus are presented in Fig. 1 (d,e,f).
Fig. 1. (a) Simulated single grating image (b) with a PMMA ball and (c) without the grating. (d,e,f) Corresponding Fourier transform modulus. The Fourier transforms of $I_{ref}$ (d) contains the 12 harmonics from the grating (denoted $H_{k,l}$ with $k,l\in \{0;\pm 1;\pm 2\}$). The Fourier transform of $I_{obj}$ (f) contains a central harmonic $H_{0,0}$ holding the absorption and edge overshoot information of the PMMA ball. Its main frequency range is wide (denoted by the yellow dotted circle). The Fourier transform of $I_{mod}$ (e) results in an overlap between the modulated $H_{k,l}$ and $H_{0,0}$. In the extraction window of $H_{-1,1}$ and $H_{1,1}$ (see red and blue square), frequencies from $H_{0,0}$ act as parasitic frequencies.
The Fourier transform of the reference interferogram (Fig. 1 (d)) contains harmonics, denoted $H_{k,l}$ with $(k,l) \in \mathbb {Z}^{2}$, since the signal $I_{ref}$ is periodic along several orientations. Therefore, it can be decomposed into a sum of sinusoidal functions oriented along the same directions. Their expression can be retrieved by performing the auto-correlation of the diffracted orders generated by the MLSI [11]. Here, the simulated grating diffracts over 5 orders leading to 12 principal harmonics in the Fourier plane: $H_{\pm 1,\pm 1}$, $H_{0,\pm 2}$, $H_{\pm 2,0}$, $H_{\pm 2,\pm 2}$, carrying redundant information of the wavefront gradient. On the other hand, the Fourier transform of the PMMA ball (Fig. 1 (f)) contains a central harmonic denoted $H_{0,0}$ holding the absorption and edge overshoot information of the object. The frequency extension of $H_{0,0}$ is wide in the Fourier plane (see the yellow circle) since the intensity of the PMMA ball evolves fast, especially at its edges. The Fourier transform of the modulated interferogram $I_{mod}$ (Fig. 1 (e)) thus results in a frequency overlap between $H_{0,0}$ and all the modulated harmonics $H_{k,l}$ of the grating.
To retrieve the phase gradient $G_{k,l}$ (Fig. 2 (a,b)), the argument of the inverse Fourier transform of $H_{k,l}$ has to be performed on an extraction window centered at the carrier frequency $f_{k,l}$ (see corresponding red and blue rectangles in Fig. 1 (e)):
Fig. 2. (a) Gradient $G_{-1,1}$ and (b) gradient $G_{1,1}$ showing artifacts at the edge of the object, region where the edge intensity overshoot evolves faster than the interferogram periodicity. (c) Phase retrieved $\phi _{1,1}$ from the gradients $G_{-1,1}$ and $G_{1,1}$ showing the influence of the artifacts from $G_{-1,1}$ and $G_{1,1}$. (d) Phase error map, defined as the difference between the theoretical phase $\phi _t$ and the retrieved phase $\phi _{1,1}$.
The size of the extraction window is given by the space between two consecutive harmonics in the Fourier plane [17]. The Fourier derivative theorem is then applied to retrieve the phase image $\phi _{k,l}$ (Fig. 2 (c)) from a couple of two orthogonal gradients $(G_{k,l}, G_{-l,k})$ [18]:
In this example, the extraction windows of $H_{k,l}$ contain the true signal and the residual signal from $H_{0,0}$, denoted $Res_{H_{0,0}}$. These parasitic frequencies lead to artifacts in the gradient estimation $G_{k,l}$. Such artifacts arise in the gradient images $G_{-1,1}$ and $G_{1,1}$, at the edge of the PMMA ball (Fig. 2 (a,b)). These are materialized by phase jumps (also called phase dislocations) and affect the final phase image (Fig. 2 (c)). A similar behavior is observed for each other phase gradient measured by the MLSI. Finally, Fig. 2 (d) displays the phase error map, that is the difference between the theoretical phase $\phi _t$ of the same PMMA ball at $17.48$ keV and the retrieved phase $\phi _{1,1}$. Here, $\phi _t=\frac {2\pi \delta _0}{\lambda _0} \Delta z$, where $\delta _0$ is the decrement value of the complex refractive index for PMMA material at the wavelength $\lambda _0$ [19] and $\Delta z$ is the projected thickness of the ball. We can observe that the errors are located at the edge of the ball, in the direction of $G_{-1,1}$ and $G_{1,1}$.
3. Reduction of artifacts in the phase retrieval process
Artifacts in phase gradient images arise as soon as frequencies from two or more harmonics overlap in the Fourier plane. The major contribution to this overlap is the central harmonic $H_{0,0}$ since its amplitude is the highest in X-ray imaging. Thus, we propose to build an interferogram $I_{corr}$ that does not hold any intensity information from the sample in order to minimize the amplitude of $H_{0,0}$. This can be done by subtracting the modulated interferogram $I_{mod}$ by the attenuation image of the sample $I_{obj}$. The new interferogram $I_{corr}$ is built according to the following expression:
where $\alpha \leq 1$ is a constant that rescales the gray values of $I_{obj}$ in order to match with $I_{mod}$; $\beta > 0$ is a constant for readjusting $I_{corr}$ around the mean of the initial interferogram $I_{mod}$. The expression of $\alpha$ and $\beta$ is given below: where $\Omega$ refers to the same region of interest (ROI) in each image, defined in an area outside the sample. Therefore, it is recommended to keep a portion of the background signal in $I_{obj}$. $<\square >_{_\Omega }$ corresponds to the mean gray value of a given intensity inside $\Omega$. The new interferogram $I_{corr}$ is then compared to the reference $I_{ref}$ and phase retrieval is performed conventionally. Figure 3 illustrates the construction of the new interferogram $I_{corr}$ and the phase retrieval process. Note that the treatment is performed in real space but can also be performed in reciprocal space since this method is linear.Fig. 3. Construction of the corrected interferogram $I_{corr}$ (see Eq. (3)) and corresponding phase retrieval process. $\Omega _i$ refers to a ROI in each image used to calculate the mean gray value of $I_{ref}$, $I_{mod}$ and $I_{obj}$ from which $\alpha$ and $\beta$ are derived.
To compare with MARIO, a classical approach is also performed. Apodization technique is used to smooth out artifacts, by filtering the highest frequencies in the extraction window [14]. Figure 4 top row shows the phase retrieval results using a Blackman window [14] and Fig. 4 bottom row shows the phase retrieval results after processing MARIO.
Fig. 4. (a) Gradient $G_{-1,1}$ and (b) gradient $G_{1,1}$ obtained after applying a Blackman window on $H_{-1,1}$ and $H_{1,1}$ respectively. (c) Associated phase image $\phi _{1,1}$ and (d) phase error map. Compared to the images shown in Fig. 2, artifacts are reduced however this process lowers the resolution of the images. (e,f,g,h) Same data after applying the proposed MARIO. Compared to the apodization technique, artifacts are efficiently reduced without affecting the resolution.
In the case of apodization, artifacts at the edge of the ball are reduced, however it reduces the resolution of the images (Fig. 4 (a,b,c)). In fact, among all frequencies filtered by the apodization window, a significant part comes from the phase modulation, that is the highest spatial frequency content of the gradient image.
In the case of MARIO, artifacts are efficiently reduced without affecting the image resolution. This leads to a better restitution of the edge of the ball in the gradients and phase images (Fig. 4 (e,f,g)). Fig. 4 (d,h) display the error maps associated to each method. We can see that the amplitude of the error map is reduced after performing MARIO (note that gray values are rescaled between (d) and (h)). In fact, it is so reduced that other local errors appear on the vertical and horizontal edges. This is due to the residual influence of neighbouring harmonics of $H_{-1,1}$ and $H_{1,1}$.
To evaluate the gain of each method, the Root Mean Square Error ($RMSE$) is performed:
4. Application on an experimental image
After demonstrating the MARIO process on a model case, we apply the method on an experimental case, the X-ray imaging of a Carbon Fiber Reinforced Polymer (CFRP). This sample, manufactured in a controlled laboratory environment serves as model for lightning damage experiment in the aeronautic field [20]. It is made of sixteen layers of aligned carbon fibers with a diameter of 12 $\pm \,3\,{\mathrm {\mu }}$m (successive layers are oriented at -45$^{\circ }$; +45$^{\circ }$) surrounded by epoxy resin. This sample is particularly well adapted to highlight the influence of fast transitions on the phase retrieval process for two reasons: central holes with very steep edges and a high resolution structure due to the carbon fibers.
The sample is imaged using the MLSI bench shown in Fig. 5. It is composed of a single 2D checkerboard grating manufactured by the Microworks company. The $\pi$-phase shift of the grating is induced by gold material of 3.49 ${\mathrm {\mu }}$m thickness deposited on a polymer substrate. The X-ray source is a divergent microfocus tube (Feinfocus FXE-160.51) of a measured spot size of 5.5 ${\mathrm {\mu }}$m with a solid transmitted tungsten anode. The detector is a high resolution sCMOS (Hamamatsu C12819-102-U) with a GadOx scintillator of 20 ${\mathrm {\mu }}$m thickness. Here, $d_{sd} = 57$ cm giving a grating magnification of $M_g=3.3$ and an object magnification of $M_o=2.3$. The grating orthogonal period and the pixel size of the detector remain unchanged from the simulation study.
Fig. 5. CFRP sample with the imaged region denoted by the red dashed line rectangle (left). Experimental bench with a microfocus X-ray source, 2D-gratings, CFRP sample and a high-resolution detector (right).
Experimental data, acquired in a specific area of the CFRP (spotted by the red dashed line rectangle in Fig. 5), are presented in Fig. 6. The left column (a,d) presents the image of the CFRP and its Fourier transform modulus. The distribution of the central harmonic $H_{0,0}$ reveals two regions: the first one is linked to the carbon fibers absorption, oriented at -45$^{\circ }$; +45$^{\circ }$ (see orange arrow in Fig. 6 (d)). Since they induce fast intensity transitions, it results in a wide distribution in the Fourier plane. The second region (see green arrow) is linked to the edge intensity overshoot of the CFRP (see zoom box in Fig. 6 (a)) and is also of wide spectral distribution.
Fig. 6. (a) Image of the CFRP sample alone $I_{obj}$. (b) Interferogram of the CFRP sample $I_{mod}$. (c) Interferogram $I_{corr}$ obtained after applying the MARIO process. Bottom row, the corresponding Fourier transform modulus. The green and yellow arrows in (d) relate to the spectral distribution of respectively the edge intensity overshoot and the absorption of the CFRP. It leads in (e) to a frequency overlap with the harmonics of the grating. In (f), the influence of the CFRP spectral distribution has been minimized according to the MARIO process.
The middle column (Fig. 6 (b,e)) presents the image of the CFRP and its associated Fourier transform modulus, acquired after placing the grating into the optical path (see zoom box in Fig. 6 (b)). The harmonics $H_{k,l}$ are distributed in the same way as for the simulation study. A frequency overlap between $H_{k,l}$ and $H_{0,0}$ is noticeable. Moreover, the amplitude of $H_{k,l}$ is lower than the amplitude of $H_{0,0}$ since the fringe intensity contrast is lower than the sample intensity contrast.
The right column (Fig. 6 (c,f)) presents the interferogram $I_{corr}$ corrected by the MARIO process and its associated Fourier transform modulus. The attenuation of the sample as well as its edge intensity overshoot has been removed, highlighted by a more uniform gray level. However, the fringe contrast inside the object stays lower than outside (see zoom box in Fig. 6 (c)). Indeed, subtracting the intensity of the object does not compensate the reduction of the fringe contrast. Therefore, in the areas where the attenuation of the object is initially the highest, the signal-to-noise ratio stays weak. In the associated Fourier transform image (Fig. 6 (f)), the amplitude of $H_{0,0}$ is successfully minimized while maintaining the amplitude of all harmonics $H_{k,l}$ unaffected.
The corrected interferogram $I_{corr}$ is then given as input in the phase retrieval process. We present the results without (top row) and with (bottom row) the MARIO process, in Fig. 7. Since the harmonics $H_{-1,1}$ and $H_{1,1}$ have the highest signal-to-noise ratio, they are used to display the phase gradients $G_{-1,1}$ and $G_{1,1}$. In the raw phase gradient images (Fig. 7 (a,b)), artifacts arise at the edges of the hole (see green arrows) and throughout the sample, evolving in the direction of the measured gradient, because of the oriented carbon fibers (-45$^{\circ }$; +45$^{\circ }$) (see orange arrows). In the phase image (Fig. 7 (c)), these artifacts induce a textured aspect of the sample where slowly evolving gray level variations give a mottled appearance. In the gradient images processed by MARIO (Fig. 7 (d,e)), edges and carbon fibers artifacts are significantly reduced, cleaning the phase image from nearly all dislocations (Fig. 7 (f)).
Fig. 7. Top row, uncorrected experimental results: (a) gradient $G_{-1,1}$ and (b) gradient $G_{1,1}$. (c) Phase image $\phi _{1,1}$ retrieved from the gradients in (a,b). The green arrows point the edge artifacts while the orange arrows point artifacts in the object. Bottom row: corresponding corrected images by the MARIO process. All artifacts are significantly reduced in the gradient and phase images.
The gain of MARIO on experimental data is evaluated by performing the Confidence Map [16]. This method gives an estimation of the experimental errors for any phase image retrieved from a gradient-based device, taking advantage of the MLSI to measure at least two orthogonal gradients simultaneously. So, it is possible to calculate a phase derivative closure map by applying the curl operator. As we assume that the wavefront issued from the sample is a continuous surface, the value of the phase derivative closure map should be equal to zero in all points. However in real conditions, this calculation differs from zero and suffers from several error contributions: noise $\epsilon _n$, under-sampling $\epsilon _u$ or dislocations $\epsilon _d$. By analyzing the histogram of the phase derivative closure map, each error type is assigned a threshold value. The Confidence Map is built by coloring the error pixels according to their type. For instance, phase dislocations are displayed with red pixels, under-sampling errors with blue pixels and noise with cyan pixels. Therefore, the Confidence Map alerts the observer to the presence of artifacts that could mislead his interpretation of the image.
Figure 8 (a) presents the Confidence Map of the CFRP raw phase image superimposed to the phase image presented in Fig. 7 (c). For better understanding, a short footage is given as supplemental materials (see Visualization 1)). $\epsilon _d$ alerts are displayed here in red, $\epsilon _u$ in blue, and $\epsilon _n$ are not displayed for better visualisation. This map is the same as the one presented in our previous work introducing the Confidence Map [16, Fig. 7 (a,b,c)]. Figure 8 (b) presents the Confidence Map of the CFRP phase image after performing MARIO; a clear reduction of the phase dislocation alerts $\epsilon _d$ as well as the under-sampling alerts $\epsilon _u$ are observed inside and at the edge of the sample (see Visualization 2). The percentage of dislocation alerts all over the phase image, defined as the ratio between the red pixel amount and the total pixel amount is equal to $\epsilon _d= 3.2\%$ before any treatment. This ratio decreases to 0.2% thanks to MARIO. In the meantime, the percentage of under-sampling alerts decreases from $\epsilon _u = 30.8\%$ to 0.7%, concluding on the significant improvement of the phase image quality. The associated gains are listed in Table 1.
Fig. 8. (a) CFRP raw phase image presented in Fig. 7 (c) superimposed to its associated Confidence Map (see Visualization 1). The under-sampling alerts $\epsilon _u$ are displayed in blue and phase dislocation alerts $\epsilon _d$ are displayed in red. (b) CFRP phase image processed by MARIO (Fig. 7 (f)) superimposed to its associated Confidence Map (see Visualization 2). A clear reduction of the alerts is noticeable inside and at the edge of the sample (see Table 1).
Table 1. Table of the Confidence Map values of the CFRP phase image before and after MARIO and associated gains.
5. Conclusion
In this paper, artifacts have been studied in phase images retrieved by Fourier demodulation technique. We demonstrate that they are mainly related to the central harmonic extension. In other words, they originate from the attenuation and from the edge intensity overshoot of the object. This observation has led to the generation of a preventive computing method, aiming at removing the parasitic frequencies from the useful ones. We call it Method of Artifacts Reduction from the Intensity of the Object (MARIO). We apply this method in the case of a Carbon Fiber Reinforced Polymer (CFRP) used in the aeronautic field and demonstrate quantitatively, thanks to the Confidence Map, that the quality of the phase image is significantly improved. Compared to an apodization technique, the proposed MARIO efficiently removes artifacts in phase images without affecting the spatial resolution. Even if this method requires to acquire an additional image, it significantly reduces the artifacts, making it a particularly suitable tool for the non-destructive testing and evaluation field, which is often interested in complex objects with abrupt transitions or high resolution texture. Finally, this method can also be applied to a wider variety of grating-based devices using Fourier demodulation technique, for instance mono or bi-directional single gratings [7,9] and Talbot-Lau interferometry technique [21].
Funding
Office National d'études et de Recherches Aérospatiales; Commissariat à l'Énergie Atomique et aux Énergies Alternatives.
Acknowledgments
The authors thanks the Department of Physics Instrumentation Environment Space (DPHY) and the Department of Materials And Structures (DMAS) from ONERA for the availability of the CFRP-test piece.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data presented in this study are available from the corresponding author on reasonable request.
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